Spatial equidistribution of binomial coefficients modulo prime powers

The spatial distribution of binomial coefficients in residue classesmodulo prime powers is studied. It is proved inter alia that empirical distributionof the points (k, m)p−m with 0 ≤ k ≤ n < pm and nk≡ a (mod p)s(for(a, p) = 1) for m → ∞ tends to the Hausdorff measure on the “p-adic Sierpi´nskigasket”, a fractals studied earlier by von Haeseler, Peitgen, and Skordev.